What counts as a standard model of arithmetic?

Question

In my research so far, I've found that the canonical standard model of arithmetic is $\mathbb{N}$ under the addition and multiplication operations. However, I've been unable to find much on any other standard model of arithmetic.

Is $\mathbb{N}$ the only standard model of arithmetic, or are there others? I intuitively imagine that $\mathbb{Q}$ under addition and $\mathbb{R}$ under addition and multiplication are standard models, but I've yet to find any support for this. Is a model either standard or nonstandard, or is there an in-between where stuff like $\mathbb{Q}$ and $\mathbb{R}$ would go? If there's more than one, then what precisely defines a standard model of arithmetic?

Answer 1

There is a standard model of arithmetic: any model of the Peano axioms + the second order axiom of induction. Any two such models are isomorphic (a fact known as the categoricity of the theory). The standard models of arithmetic are precisely these models.

A similar story holds for the reals. Any model of the axioms of a complete ordered field is a standard model of the reals. Any such models are isomorphic, so again the theory is categorical.

In light of this, the obtain nonstandard models one must relax some of the axioms (for each theory). Nonstandard models of arithmetic are basically models of the Peano axoims + the induction scheme axioms (one axiom for every relevant sentence, so countably many axioms, but all are first order). Any such model that is not isomorphic to the standard model of arithmetic is called a nonstandard model. There are plenty such models just by virtue of the Lowenheim-Skolem theorem + the compactness theore: There is a nonstandard model of any infinite cardinality. So there are plenty of non-isomorphic models of nonstandard arithmetic.

A similar story holds for nonstandard models of the reals. It's also worth mentioning that models can also be constructed using ultrapowers and all constructions rely on some form of the axiom of choice. Also, historically, nonstandard models appeared in logic via the Lowenheim-Skolem theorem, and made their serious appearance in (more) mainstream (at least at that time) mathematics in the work of Robinson on nonstandard analysis.

Answer 2

The ordinary natural numbers, under the usual addition and multiplication, are a standard model of arithmetic. Any model not isomorphic to these is called non-standard. The usual convention in logic is that by $\mathbb{N}$ we mean the numbers $0,1,2,\dots$.

We have left the term arithmetic undefined. One usually has a particular theory in mind, such as first order Peano arithmetic. Or perhaps the theory whose axioms are all sentences of the usual language of arithmetic that are true in $\mathbb{N}$. That is a theory incomparably stronger than first order Peano arithmetic.

The term arithmetic may be confusing here. The word has many meanings. In particular, it is an old-fashioned term for number theory. That is what arithmetic means when we refer to non-standard models of arithmetic.

Remark: The structures $\mathbb{Q}$ and $\mathbb{R}$, under the usual operations, are not models of arithmetic. To see this, let $\varphi$ be the sentence $\forall x(\lnot(x=0)\longrightarrow \exists y(xy=1))$. Then $\varphi$ is false in $\mathbb{N}$, but true in $\mathbb{Q}$ and $\mathbb{R}$. There are many other similar sentences.